Condorcet’s Paradox and Social Preferences

This lecture shows a major problem with aggregating social preferences.

Takeaway Points

Suppose we have a group of individuals with rational (complete and transitive) preferences. Many seemingly sensible rules for aggregating these preferences can result in irrational social preferences. Majority rule is one of these.

For example, suppose Marco prefers A to B to C, Ted prefers B to C to A, and Donald prefers C to A to B. Consider how majority rule would determine the group’s preference between A and B. Marco and Donald prefer A to B, and thus majority rule means the group prefers A to B. Now consider the group’s preference between B and C. Marco and Ted prefer B to C, and thus majority rule means the group prefers B to C. Finally, consider the group’s preference between A and C. Ted and Donald prefer C to A, and thus majority rule means that the group prefers C to A.

Combining these preferences together, the group prefers A to B, B to C, and C to A. Despite no individual having a preference cycle, their collective preference cycles!

Marquis de Condorcet was the first person to point out this problem with majority rule. We therefore call the disconnect between individual and social preferences Condorcet’s paradox.

Other methods of preference aggregation can sidestep the paradox. However, all other solutions come with problems of their own. See Arrow’s impossibility theorem.

We consequently need to be careful about modeling group preferences in a game theoretical context.